I have the 1956 movie Around the World in 80 Days on DVD. In fact, I have it on 2 DVDs. Half the movie is on the first DVD and the second half is on the second DVD. I'd like to make a backup of this movie to one Xvid AVI file if possible.

Should I have any concerns about using VOBMerge to merge the VOBs from the two discs, before then encoding the result to AVI? Does it have any impact to merge VOBs that are not the same bitrate as each other (which is the case here)?

Normally the IFO file gets loaded in Auto Gordian Knot. If I merge the two VOB files, should I just load the merged VOB into AutoGK and forget about the IFO files?

You should "recreate" a movie, not simply joining some VOBs. You can do this with many tools, like pgcedit, vobblanker (despite its name) etc etc. Then send the whole movie to your encoder to xvid. Xvid is however very old and makesno real sense to still use it.
Oh yes, there is no problem to have different bitrates in each half.

Hmmm...I can see no reason why you have to merge the VOBs prior encoding.
Simply ensure that you have the same settings for both encodes and do each DVD separately to your target format of choice.
Then simply merge the two resulting files with AVIDemux or VirtualDub in stream copy. Usually you do not see any visual distortion at the merge point. I have also merged a lot of two CD DivX from the days back then to a single file like this and never had issues.

Edit:
Another more simplistic approach might be simply merging all VOBs from both DVDs binary ("copy /B" under DOS) and then feed the video to AutoGK.

In regard to the format I agree with Ghitulescu: XVid is not up to date anymore. If possible I would rather use h264 which is more suitable for older movies and keeps more details. If I was in your position I would go for Handbrake and use at least some medium denoiser.